Given a discrete sample of event locations, we wish to produce a probability density that models the relative probability of events occurring in a spatial domain. Standard density estimation techniques do not incorporate priors informed by spatial data. Such methods can result in assigning significant positive probability to locations where events cannot realistically occur. In particular, when modelling residential burglaries, standard density estimation can predict residential burglaries occurring where there are no residences. Incorporating the spatial data can inform the valid region for the density. When modelling very few events, additional priors can help to correctly fill in the gaps. Learning and enforcing correlation between spatial data and event data can yield better estimates from fewer events. We propose a non-local version of maximum penalized likelihood estimation based on the H(1) Sobolev seminorm regularizer that computes non-local weights from spatial data to obtain more spatially accurate density estimates. We evaluate this method in application to a residential burglary dataset from San Fernando Valley with the non-local weights informed by housing data or a satellite image.

The concentration of police resources in stable crime hotspots has proven effective in reducing crime, but the extent to which police can disrupt dynamically changing crime hotspots is unknown. Police must be able to anticipate the future location of dynamic hotspots to disrupt them. Here we report results of two randomized controlled trials of near real-time epidemic-type aftershock sequence (ETAS) crime forecasting, one trial within three divisions of the Los Angeles Police Department and the other trial within two divisions of the Kent Police Department (United Kingdom). We investigate the extent to which (i) ETAS models of short-term crime risk outperform existing best practice of hotspot maps produced by dedicated crime analysts, (ii) police officers in the field can dynamically patrol predicted hotspots given limited resources, and (iii) crime can be reduced by predictive policing algorithms under realistic law enforcement resource constraints. While previous hotspot policing experiments fix treatment and control hotspots throughout the experimental period, we use a novel experimental design to allow treatment and control hotspots to change dynamically over the course of the experiment. Our results show that ETAS models predict 1.4–2.2 times as much crime compared to a dedicated crime analyst using existing criminal intelligence and hotspot mapping practice. Police patrols using ETAS forecasts led to an average 7.4% reduction in crime volume as a function of patrol time, whereas patrols based upon analyst predictions showed no significant effect. Dynamic police patrol in response to ETAS crime forecasts can disrupt opportunities for crime and lead to real crime reductions.